Stability,Bifurcation And Simulation Of Two Discrete Predator-prey Systems

The study of the dynamical properties of discrete predator-prey systems is one of the important research topics in biomathematics.This thesis mainly studies the stability and bifurcation problems of two types of discrete systems.The whole thesis consists of five chapters.The main contents are as follows:In Chapter 1,we provide an overview of the research background,the significance of the research and main work done in this thesis.In Chapter 2,we briefly review some basic concepts about discrete systems,the central manifold theorem and local bifurcation theory.In Chapter 3,by using the forward Euler method,we derive a discrete predator-prey system of Gause type with constant-yield prey harvesting and a monotonically increasing functional response.Firstly,we use an important lemma to study in detail the existence and local stability of fixed points.Afterwards,through the central mani-fold theorem and the bifurcation theory,we obtain sufficient conditions for saddle-node bifurcation and flip bifurcation of the system to occur.Finally,the numerical simula-tions are carried out by using Matlab software,which illustrate the theoretical results and reveals some new dynamical phenomena of system–chaos.One of the foci of this chapter is to skillfully find a reversible transformation to derive the standard form of the flip bifurcation of the system,and determine the stability of closed orbit bifurcated,which cannot be obtained by routine methods because its two characteristic roots are multiple roots,so the corresponding reversible matrix does not exist.In Chapter 4,we study a discrete predator-prey system of ratio-dependent func-tional response,and completely proves the existence and stability of three non-negative equilibria E₁,E₂and E₃,given the parameters.Thereafter,using the central manifold theorem and the bifurcation theory,we obtain the conditions for the occurrences of transcritical bifurcation and Neimark-Sacker bifurcation.Finally,the existence of the Neimark-Sacker bifurcation is illustrated by numerical simulations.